|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > sbcieg | Structured version Visualization version GIF version | ||
| Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) Avoid ax-10 2140, ax-12 2176. (Revised by GG, 12-Oct-2024.) | 
| Ref | Expression | 
|---|---|
| sbcieg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| sbcieg | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-sbc 3788 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
| 2 | sbcieg.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 3 | 2 | elabg 3675 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) | 
| 4 | 1, 3 | bitrid 283 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 {cab 2713 [wsbc 3787 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-sbc 3788 | 
| This theorem is referenced by: sbcie 3829 2nreu 4443 reuprg0 4701 rabsnif 4722 ralrnmptw 7113 ralrnmpt 7115 fpwwe2lem3 10674 nn1suc 12289 opfi1uzind 14551 mndind 18842 fgcl 23887 cfinfil 23902 csdfil 23903 supfil 23904 fin1aufil 23941 ifeqeqx 32556 nn0min 32823 bnj1452 35067 cdlemk35s 40940 cdlemk39s 40942 cdlemk42 40944 2nn0ind 42962 zindbi 42963 prproropreud 47501 | 
| Copyright terms: Public domain | W3C validator |